LIFTINGS OF 1 - FORMS TO THE LINEAR r - TANGENT BUNDLE

Let r; n be xed natural numbers. We prove that for n-manifolds the set of all linear natural operators T ! T T (r) is a nitely dimensional vector space over R. We construct explicitly the bases of the vector spaces. As a corollary we nd all linear natural operators T ! T r. All manifolds and maps are assumed to be innnitely diierentiable. 0. Let r; n be xed natural numbers. Given a manifold M we denote by T r M = J r (M; R) 0 the space of all r-jets of maps M ! R with target 0. This is a vector bundle over M with the source projection. The dual vector bundle (T r M) of T r M is denoted by T (r) M and called the linear r-tangent bundle of M. We denote the bre of T r M and T (r) M over x by T r x M and T (r) x M respectively. Every embedding ' : M ! N of two n-manifolds induces a vector bundle homomorphism T r ' : T r M ! T r N over ' deened by T r '(j r x) = j r '(x) (' ?1) for any : M ! R and any x 2 M with (x) = 0, where by j r x we denote the r-jet of at x. This embedding induces also a vector bundle homomorphism T (r) ' : T (r) M ! T (r) N over ' dual to T r ' ?1 , i.e. T (r) '(()(j r '(x)) = (j r x (')) for any 2 T (r) x M, any x 2 M and any j r '(x) 2 T r '(x) N, cf. 4]. In this paper we study the problem how a 1-form ! on a manifold M can induce a 1-form on T (r) M and a section of T r M ! M. This problem is reeected in the concept of linear natural operators T ! T T (r) and T ! T r , cf. 4]. In the fundamental monograph 4] there is a very general deenition of natural operators. We restrict ourselves to the following one. Deenition 0.1. Let r; n be xed natural numbers. Let 1 (M) denotes the vector space of all 1-forms on M and ?T r M denotes the vector space of all …